Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.
By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.
\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.
\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).
\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.
The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.
\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.
Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv
Source: https://github.com/pcm-dpc/COVID-19
url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
# DT::datatable(COVID19)Warnings
- 15/04/2020: dati Regione Friuli Venezia Giulia ricalcolati (ricalcolo di isolamento domiciliare e dimessi/guariti)
- 12/04/2020: dati P.A. Bolzano ricalcolati (ricalcolo dati guariti -110 rispetto a ieri).
- 10/04/2020: dati Regione Molise parziali (dato tamponi non aggiornato).
- 29/03/2020: dati Regione Emilia Romagna parziali (dato tampone non aggiornato).
- 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
- 18/03/2020: dati Regione Campania non pervenuti.
- 18/03/2020: dati Provincia di Parma non pervenuti.
- 17/03/2020: dati Provincia di Rimini non aggiornati
- 16/03/2020: dati P.A. Trento e Puglia non pervenuti.
- 11/03/2020: dati Regione Abruzzo non pervenuti.
- 10/03/2020: dati Regione Lombardia parziali.
- 07/03/2020: dati Brescia +300 esiti positivi
# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$totale_casi,
dy = reldiff(COVID19$totale_casi))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 15711.525711 1868.962382 8.407 0.0000000000166 ***
## th2 0.045419 0.002435 18.654 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18360 on 56 degrees of freedom
##
## Number of iterations to convergence: 12
## Achieved convergence tolerance: 0.000007523mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 184754.7175 2407.9476 76.73 <2e-16 ***
## xmid 34.8392 0.3173 109.80 <2e-16 ***
## scal 7.7557 0.2128 36.45 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3957 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000005336mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 1000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 215331.1799185 1815.1380116 118.63 <2e-16 ***
## b2 8.8710489 0.1926063 46.06 <2e-16 ***
## b3 0.9342153 0.0008802 1061.36 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1410 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000006901richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "plinear",
control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging...
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 226239.498420 2463.559130 91.83 <2e-16 ***
## th2 0.058334 0.001135 51.40 <2e-16 ***
## th3 6.247310 0.181305 34.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1357 on 55 degrees of freedom
##
## Number of iterations to convergence: 2
## Achieved convergence tolerance: 0.07077
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3,
# data = data, start = start, trace = TRUE)models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -650.7313 | 3 | 0.9300571 | 1307.463 | 1307.907 | 1313.644 | |
| Logistic model | -561.1916 | 4 | 0.9968771 | 1130.383 | 1131.138 | 1138.625 | |
| Gompertz model | -501.3350 | 4 | 0.9995433 | 1010.670 | 1011.425 | 1018.912 | |
| Richards model | -499.0977 | 4 | 0.9996007 | 1006.195 | 1006.950 | 1014.437 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Infected", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 10000),
minor_breaks = seq(0, coef(mod2)[1], by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(100,NA)) +
labs(y = "Infected (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
minor_breaks = seq(0, max(ylim), by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 59 2020-04-22 229087 178317 280174
## 591 2020-04-22 176906 167218 184458
## 592 2020-04-22 183478 180062 186982
## 593 2020-04-22 184629 181485 188203
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$deceduti,
dy = reldiff(COVID19$deceduti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 1468.47695 194.50573 7.55 0.000000000427 ***
## th2 0.05152 0.00266 19.37 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2288 on 56 degrees of freedom
##
## Number of iterations to convergence: 13
## Achieved convergence tolerance: 0.000003477mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 24859.7877 329.6510 75.41 <2e-16 ***
## xmid 37.5953 0.2930 128.30 <2e-16 ***
## scal 7.2087 0.1901 37.93 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 480.9 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000007466mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 10000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 29581.0907685 252.1905181 117.30 <2e-16 ***
## b2 12.3767926 0.2963853 41.76 <2e-16 ***
## b3 0.9313193 0.0008655 1076.09 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 163.3 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000003948richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 30667.228995 313.169040 97.92 <2e-16 ***
## th2 0.064125 0.001079 59.43 <2e-16 ***
## th3 9.446696 0.283373 33.34 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 158.2 on 55 degrees of freedom
##
## Number of iterations to convergence: 5
## Achieved convergence tolerance: 0.000005552models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -529.9416 | 3 | 0.9404290 | 1065.8832 | 1066.3277 | 1072.0646 | |
| Logistic model | -438.9426 | 4 | 0.9974492 | 885.8853 | 886.6400 | 894.1270 | |
| Gompertz model | -376.2945 | 4 | 0.9996565 | 760.5891 | 761.3438 | 768.8308 | |
| Richards model | -374.4468 | 4 | 0.9996863 | 756.8936 | 757.6483 | 765.1354 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Deceased", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Deceased (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 59 2020-04-22 30687 24491 37197
## 591 2020-04-22 23646 22381 24535
## 592 2020-04-22 24561 24160 24972
## 593 2020-04-22 24676 24262 25094
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$dimessi_guariti,
dy = reldiff(COVID19$dimessi_guariti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 1309.429143 119.892672 10.92 1.7e-15 ***
## th2 0.064700 0.001783 36.29 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2171 on 56 degrees of freedom
##
## Number of iterations to convergence: 11
## Achieved convergence tolerance: 0.000009039mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 69117.9931 2438.2017 28.35 <2e-16 ***
## xmid 49.1937 0.7311 67.29 <2e-16 ***
## scal 9.3531 0.2501 37.40 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 802.8 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000001658mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 150396.825782 8258.863484 18.21 <2e-16 ***
## b2 8.422957 0.150164 56.09 <2e-16 ***
## b3 0.965333 0.001064 907.55 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 415.1 on 55 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000004762richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, # algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 333921.576942 47788.090085 6.988 3.95e-09 ***
## th2 0.017099 0.001519 11.253 6.87e-16 ***
## th3 4.054451 0.153347 26.440 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 345.8 on 55 degrees of freedom
##
## Number of iterations to convergence: 19
## Achieved convergence tolerance: 0.000004128models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -526.8876 | 3 | 0.9851402 | 1059.7751 | 1060.2196 | 1065.9565 | |
| Logistic model | -468.6690 | 4 | 0.9978460 | 945.3379 | 946.0926 | 953.5797 | |
| Gompertz model | -430.4098 | 4 | 0.9993470 | 868.8195 | 869.5742 | 877.0613 | |
| Richards model | -419.8230 | 4 | 0.9995318 | 847.6460 | 848.4007 | 855.8878 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Recovered", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Recovered (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 59 2020-04-22 59554 52992 66383
## 591 2020-04-22 51180 48746 53257
## 592 2020-04-22 52602 51520 53548
## 593 2020-04-22 53086 52190 53882
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19$data,
positives = c(NA, diff(COVID19$totale_casi)),
swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
# df$y = df$positives/df$swabs
df$y = df$positives/c(NA, zoo::rollmean(df$swabs, 2))
df = subset(df, swabs > 50)
# DT::datatable(df[,-4], )ggplot(df, aes(x = date)) +
geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
geom_line(aes(y = swabs, color = "swabs")) +
geom_point(aes(y = positives, color = "positives"), pch = 0) +
geom_line(aes(y = positives, color = "positives")) +
labs(x = "", y = "Number of cases", color = "") +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = palette()[c(2,1)]) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))ggplot(df, aes(x = date, y = y)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col=palette()[4]) +
geom_line(size = 0.5, col=palette()[4]) +
labs(x = "", y = "% positives among admnistered swabs (two-day rolling mean)") +
scale_y_continuous(labels = scales::percent_format(),
breaks = seq(0, 0.5, by = 0.05)) +
coord_cartesian(ylim = c(0,max(df$y, na.rm = TRUE))) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19$data,
hospital = c(NA, diff(COVID19$totale_ospedalizzati)),
icu = c(NA, diff(COVID19$terapia_intensiva)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1ggplot(df, aes(x = date, y = hospital)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col = "orange") +
geom_line(size = 0.5, col = "orange") +
labs(x = "", y = "Change hospitalized patients") +
coord_cartesian(ylim = range(df$hospital, na.rm = TRUE)) +
scale_y_continuous(minor_breaks = seq(min(df$hospital, na.rm = TRUE),
max(df$hospital, na.rm = TRUE),
by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))ggplot(df, aes(x = date, y = icu)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col = "red2") +
geom_line(size = 0.5, col = "red2") +
labs(x = "", y = "Change ICU patients") +
coord_cartesian(ylim = range(df$icu, na.rm = TRUE)) +
scale_y_continuous(minor_breaks = seq(min(df$icu, na.rm = TRUE),
max(df$icu, na.rm = TRUE),
by = 10)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))